Differential equation question (HELP)

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Hi

Given the non-homogenous higher order differential equation

[tex]u'' - 2 u' - 8 u = 6 - 8t[/tex]

with the intial conditions u(1) = 0 and u'(1) = 1.

Find the solution.

I'm told by my professor that I'm supposed to use the fact

[tex]{u''} + {a_1} u' + {a_0} u = f [/tex]
The characteristic polynomial for this differential equation is supposedly

[tex]z^2 + {a_1} z + {a_0} z = 0[/tex]

Using this fact I find root of the polynomial of the differential equation above to be [tex]\lambda _{1,2} = 4,-2[/tex]

The I'm suppose to use the fact that (Existence and Uniqueness theorem of Non-liniear differential equations) if [tex]\lambda {_1} \neq \lambda {_2}[/tex]

[tex]u_0 (t) : = \int \limit_{t_0} ^t \frac{e^{\lambda_1 (t-s)} - e^{\lambda_2 (t-s)}}{\lambda_1 - \lambda_2} f(s) ds[/tex]

[tex]t \in I[/tex]

My solution.

Since lambda 1 and lambda 2 er different then there exist a unique partiqular solution in the form:

[tex]u(t) = C_1 e^{6-8t} + C_2 e^{-8} = 0[/tex]

By inserting part 1 of the condition into the above, I get that C_1 = C_2 = 0.

initial condition 1 = true.

I differentiate u(t) and get [tex]u'(t) = -8 e^{-2} C_1 = 1[/tex]

here I get that [tex]C_1 = \frac{1}{-8{e^{-2}}} = \frac{-e^2}{8}[/tex] and this proves that u'(1) = 1

Am I on the right track here now?

Sincerley Yours
Fred
 
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Answers and Replies

  • #2
arildno
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That would be the solution of the associated HOMOGENOUS problem (i.e, when f=0).
In addition to that, you need a particular solution of your diff. eq, try with the trial function at+b, where a, b are constants to be determined
 
  • #3
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Please look at my corrected solution above. Am I on the right track?

Sincerely
Fred
 
  • #4
arildno
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Your u(t) is NOT a solution of your original diff eq; it is part of it!
 
  • #5
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arildno said:
Your u(t) is NOT a solution of your original diff eq; it is part of it!

Okay the I not sure but then full solution is

u(t) = C_1 e^{6-8t} + C_2 e^{-8} = 6 - 8t ??

Sincerley

Fred
 
  • #6
arildno
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No, insert in your diff. eq. the trial function:
[tex]u(t)=C_{1}e^{\lambda_{1}t}+C_{2}e^{\lambda_{2}t}+at+b[/tex]
What equations do you see that "a" and "b" must fulfill in order for u(t) to be a solution?
 
  • #7
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arildno said:
No, insert in your diff. eq. the trial function:
[tex]u(t)=C_{1}e^{\lambda_{1}t}+C_{2}e^{\lambda_{2}t}+at+b[/tex]
What equations do you see that "a" and "b" must fulfill in order for u(t) to be a solution?

a and b must have such values that the solution is equal to the original differential equation ?

/Fred
 
  • #8
arildno
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Almost:
a and b must have such values that the trial function is a SOLUTION of the original differential equation.
 
  • #9
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by inserting a trail equation u(t) = at^2 + b

I get a = 0 and b = 2.

Does that sound right?

Sincerely Fred
 
  • #10
arildno
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No, you should insert at+b:
the second derivative of this is zero, so you end up with the equation for a, b:
[tex]-2a-8(at+b)=6-8t[/tex]
Thus, we get a=1 to match the first power in t, and to zeroth order in t, we're left with:
-2*1-8b=6, that is b=-1
 
  • #11
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Okay then the solution is

[tex]u(t)=C_{1}e^{\lambda_{1}t}+C_{2}e^{\lambda_{2}t}+ t - 1[/tex]

But what about the lambda values could you give me a hint on how til determain them?

Sincerely

Fred
 
  • #12
arildno
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Well, you have already found them as the solutions of the characteristic equation!
Surely you have seen this type of exercises before?
 
  • #13
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Yes of course, I have had the flue this last week, so I have been under the weather ;)

Just to recap lambda values are found as the solution of

u(t) = z(t)^2 - 2 z(t) - 8 = 0 ?

Sincerely Fred.

p.s. Talking about existence and uniquness. The exist only one solution for the differential equation. Since the solution polynomial for diff. equation is unique ?
 
  • #14
arildno
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Yes.
You now find the the UNIQUE solution of the whole initial value problem by fitting the C's so that the initial conditions are satisfied.
 

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